MAT342/CPS342 Numerical Analysis Spring 2017
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1 MAT342/CPS342 Numerical Analysis Spring 2017 General Information Meeting Time and Place Monday, Wednesday, and Friday: 2:10 3:10 p.m., KOS 126. Professor Dr. Jonathan Senning, 246 Ken Olsen Science Center , Office Hours Monday, Wednesday, Friday: 3:20 4:20 p.m., Tuesday and Thursday: 1:30 3:00 p.m., and by appointment. Text Numerical Mathematics and Computing, 7th Edition, W. Cheney and D. Kincaid, Brooks/Cole, Prerequisite Satisfactory completion of two semesters of calculus and the ability to write simple computer programs with conditionals and looping structures. Students will benefit from already having taken MAT225 Differential Equations and/or MAT232 Linear Algebra. Online Materials Online materials associated with this class can be found on the departmental web server at Grades will be maintained using Blackboard. Students with Disabilities Gordon College is committed to assisting students with documented disabilities. (See Academic Catalog Appendix C, for documentation guidelines.) A student with a disability who may need academic accommodations should follow this procedure: 1. Meet with a staff person from the Academic Support Center (Jenks 412, x4746) to: a) make sure documentation of your disability is on file in the ASC, b) discuss the accommodations for which you are eligible, c) discuss the procedures for obtaining the accommodations, and d) obtain a Faculty Notification Form. 2. Deliver a Faculty Notification Form to each course professor within the first full week of the semester; at that time make an appointment to discuss your needs with each professor. Last Modified: :19:58 AM 1
2 Failure to register in time with your professor and the ASC may compromise our ability to provide the accommodations. Questions or disputes about accommodations should be immediately referred to the Academic Support Center. See Grievance Procedures available from the ASC. Course Description Introduction In many disciplines, certainly in the natural sciences and many social sciences, there exist problems which can be modeled mathematically such that solving the mathematical model requires the solution of differential equations. Aside from simple textbook problems, most of these differential equations must be solved numerically, often by reducing the problem to that of solving one or more linear systems. Other problems also arise frequently: finding the solutions to single variable and multivariable equations, fitting functions to data, generating random numbers, etc. While the numerical problems we survey in this course are fairly simple, all of topics we will discuss are used in various ways in the solution of much more complicated problems. Some of the applications are easy to see and will be pointed out to you; others are more subtle. By now you know that what comes out of a computer is only as good as what goes in; actually it might be better to say that what comes out of a computer is no better than what goes in. Other difficulties can arise as well. We will see that it is not good enough to simply solve a mathematical problem, we also must justify, to ourselves at the very least but usually to whoever we are solving the problem for, that our answer is correct with a certain degree of accuracy. Course Content and Objectives This course will cover most of Chapters 1 12: Chapter 1: Mathematical Preliminaries and Floating-Point Representation Chapter 2: Linear Systems Chapter 3: Nonlinear Equations Chapter 4: Interpolation and Numerical Differentiation Chapter 5: Numerical Integration Chapter 6: Spline Functions Chapter 7: Initial Value Problems Chapter 8: More on Linear Systems Chapter 9: Least Squares Methods and Fourier Series Chapter 10: Monte Carlo Methods and Simulation Chapter 11: Boundary Value Problems Chapter 12: Partial Differential Equations Students completing this course will be able to effectively use computational tools to solve a variety of mathematical problems, specifically those listed above, understand and avoid potential sources of error in numerical methods, and quantify the possible error in numerical solutions. 2
3 Procedure and Expectations Class time will primarily be devoted to lecture and discussion. I encourage you to ask questions during class regarding the material presented and at times I may ask you to perform some work during our class meeting times. For each semester hour of credit, students should expect to spend a minimum of 2 3 hours per week outside of class in engaged academic time. This time includes reading, writing, studying, completing assignments, lab work, or group projects, among other activities. I expect that during class you will not use your cell phone, tablet or laptop for non-class related conversations or activities. These activities prevent you from fully concentrating on our topic and they are often distracting to those around you. Course Requirements Attendance You are expected to attend class and will be responsible for what transpires in class regardless of your attendance. As a courtesy to others, please avoid arriving late and do not leave during class unless it is an emergency or you have made prior arrangements with me. If you are aware of classes you will need to miss because of field trips, athletic events, or for personal reasons, please let me know in advance. Homework Assignments Homework assignments will be assigned throughout the semester and will usually be due the following class period. Late homework will generally not be accepted except in unusual circumstances or by prior arrangement with the instructor. You are expected to attempt all the assigned problems before the next class period. The following are required for all assignments: Assignments are to be done on 8.5x11 paper. Pages must not have ragged edges from spiral bound notebooks. Solutions should be laid out in an organized, legible manner and presented in the order that they were assigned. Final answers (where appropriate) should be clearly marked, either by highlighting or by enclosing in a neat circle or box. Multiple page assignments must be fastened (e.g. stapled but not folded) together. You are permitted to work together on the homework assignments. However, the work you turn in should be your own. These problems should be considered tools to help you better understand the theory and to become more proficient with the techniques of this course. It is essential that you understand the solution to each problem in order to derive the greatest benefit from this course. Computer Problems Throughout the semester you will be problems that require the use of a computer and frequently will involve small amounts of programming. These will vary in complexity and are designed to give you some hands-on experience with the concepts and methods discussed in the class. While all of these exercises could be carried out using programming languages such as C++ or Java, we will use the high- 3
4 level languages Python ( Octave ( or MATLAB. These interpreted languages have the advantage that one can issue commands interactively or group commands together to form a program. We will use these languages to "tinker" with the topics we are working on and also to create entire programs designed to solve specific problems. Term Project Each student will carry out a research project during the semester. These will be done in teams of two or individually. Ideally this project will allow you to explore an area where numerical computing overlaps another area of interest that you have. You will need to choose a topic and discuss it with me to get my approval during the first four weeks of the semester. These projects should involve programming and numerical experiments as well as have a significant writing component. You will need to locate and utilize references beyond our text, carry out your experimentation and then write a report that explains your work in a coherent manner. The projects will be due two weeks before the end of the semester to allow me time to comment on them and return them to you for revisions. The final version will be due at the end of the semester. Examinations There will be a midterm exam and a comprehensive take-home final exam. Any missed exam will receive the grade of zero. You may petition me in writing if you must miss an exam. This should be done as soon as possible, preferably before the exam. If I agree that you have a valid reason to miss the exam during the scheduled hour, I will arrange an alternative time for you to take the exam. Grading Procedure Your final average will be computed using the following table: Component Percentage Homework 20% Computer Exercises 20% Term Project 20% Midterm 20% Final 20% The following table shows the correspondence between the final average and letter grades. (100 96] A+ (88 84] B+ (76 72] C+ (64 60] D+ (96 92] A (84 80] B (72 68] C (60 56] D (92 88] A (80 76] B (68 64] C (56 52] D 4
5 Tentative Schedule Date Section(s) Topic Project Jan 20, Fri 1.1 Introduction Jan 23, Mon 1.2 Mathematical Preliminaries: Taylor Series Jan 25, Wed 1.3 Floating-Point Representation Jan 27, Fri 1.4 Loss of Significance Jan 30, Mon 2.1 Naïve Gaussian Elimination Feb 1, Wed 2.2 Gaussian Elimination with Scaled Partial Pivoting Feb 3, Fri 2.3 Tridiagonal and Banded Systems Feb 6, Mon 3.1 Bisection Method Feb 8, Wed 3.2 Newton's Method Feb 10, Fri 3.3 Secant Method Feb 13, Mon 4.1 Polynomial Interpolation Init. Topic Feb 15, Wed 4.2 Errors in Polynomial Interpolation Feb 17, Fri 4.3 Estimating Derivatives and Richardson Extrapolation Feb 20, Mon 5.1 Numerical Integration Feb 22, Wed 5.1 Trapezoid Method Feb 24, Fri 5.2 Romberg Algorithm Feb 27, Mon 5.3 Simpson's Rule and Adaptive Simpson's Rule Mar 1, Wed 5.4 Gaussian Quadrature Formulas Final Topic Mar 3, Fri 6.1 First-Degree and Second-Degree Splines Mar 6, Mon 6.2 Natural Cubic Splines Mar 8, Wed Midterm Exam: Chapters 1 6 Mar 10, Fri No Class Quad Finals Mar 13, Mon No Class Spring Break Mar 15, Wed No Class Spring Break Mar 17, Fri No Class Spring Break Mar 20, Mon 7.1 Taylor Series Methods Mar 22, Wed 7.2 Runge-Kutta Methods Mar 24, Fri 7.3 Adaptive Runge-Kutta Method Mar 27, Mon 7.3 Multistep Methods Outline due Mar 29, Wed 7.4 Methods for First and Higher Order Systems Mar 31, Fri 8.1 Matrix Factorizations Apr 3, Mon 8.4 Iterative Solutions of Linear Systems Apr 5, Wed 9.1 Method of Least Squares Apr 7, Fri 9.2 Orthogonal Systems and Chebyshev Polynomials Apr 10, Mon 9.3 Other Examples of the Least-Squares Principle Apr 12, Wed 10.1 Random Numbers Apr 14, Fri No Class Good Friday Apr 17, Mon No Class Easter Travel Apr 19, Wed 10.2 Est. of Areas and Volumes by Monte Carlo Techniques Apr 21, Fri 10.3 Simulation 5
6 Date Section(s) Topic Project Apr 24, Mon 11.1 Shooting Method Apr 26, Wed 11.2 A Discretization Method Paper due Apr 28, Fri Intro to 12 Partial Differential Equations and Finite Differences May 1, Mon 12.1 Parabolic Problems May 3, Wed 12.2 Hyperbolic Problems May 5, Fri 12.3 Elliptic Problems May 8, Mon Student Presentations May 10, Wed Student Presentations May 17, Wed Final exam due: 2:00 p.m. Revision due 6
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